As the 2014 Sochi Games wind down, and the traditional torrent of post-Olympics analysis, excuses, recriminations and celebrations are poised to begin, LIFE looks back 50 years to another Winter Games, in Innsbruck in 1964. It was a vastly different time, of course: the Cold War was at its most intense (the USSR won the most medals and the most golds); athletes weren’t tweeting or Instagraming their every move; 36 countries participated, fewer than half of the 88 at Sochi. And yet — some elemental things about the Olympics remain the same.

Here, in photos made by LIFE’s Ralph Crane, George Silk and Paul Schutzer, we see the same intensity in the athlete’s faces, the same striving for excellence, that we see every four years in both the summer and winter Olympiads. We see the same spirit of togetherness that seems to win out — however briefly — over the constant drumbeat of nationalism. In short, we see many of the same familiar, comforting scenes that greet us every time the Olympics roll around. And maybe that’s why, in the end, we like the Olympics so much: because in the midst of all the drama about who will win what event, and by what margin, so much about the Games remains the same.